I have a space $\mathcal{F}$ of vector-valued functions $f\in\mathcal{F}:\mathbb{R}^d \rightarrow \mathbb{R}^n$. Consider a function $\Pi: \mathcal{F} \rightarrow \mathcal{F}$:
$$ \Pi (f) (x) = \sum_{i=1}^{P} K \left(x,x_i\right) f(x_i) $$
where $x\in\mathbb{R}^d$, and $K:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^{n\times n}$. To clarify, $\Pi (f) (x)$ means "evaluate $\Pi (f) \in \mathcal{F}$ at $x$".
Our dynamical equation for a function $f$ is:
$$ \frac{\partial f_t}{\partial t}=\Pi(f_t) $$
$t$ is a time in $\mathbb{R}$. Apparently, the solution is
$$ f_t = \exp(t\Pi) f_0$$
but I am not sure how to prove this. The solution for a matrix differential equation has the same form and I can prove it for that, but I am not sure how to do it for this functional case. If possible, I would like to prove this without using matrix notation (i.e. approximating a function as a vector).
To provide the context, this is from the neural tangent kernel paper by Jacot et al.